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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Classification of skew symmetric matrices

Author: Berndt Brenken
Journal: Proc. Amer. Math. Soc. 108 (1990), 163-169
MSC: Primary 15A72; Secondary 15A21
MathSciNet review: 986646
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Abstract: The group ${\text {GL(}}d,\mathbb {Z}{\text {) = Aut(}}{\mathbb {Z}^d}{\text {)}}$ acts on the $\mathbb {Z}$-module $\operatorname {Hom} {\text {(}}{\Lambda ^2}{\mathbb {Z}^d},\mathbb {Z}/a\mathbb {Z}){\text {by}}\varphi \to \varphi {\text {(}}\alpha \Lambda \alpha {\text {)}}\quad {\text {(}}\alpha \in {\text {Aut}}{\mathbb {Z}^d}{\text {)}}$. Associated with each $\varphi$ in $\operatorname {Hom} {\text {(}}{\Lambda ^2}{\mathbb {Z}^d},\mathbb {Z}/a\mathbb {Z})$ is a finite set of invariants completely describing the orbit of $\varphi$ under this action. The result holds with $\mathbb {Z}$ replaced by an arbitrary commutative principal ideal domain.

References [Enhancements On Off] (What's this?)

  • Berndt Brenken, A classification of some noncommutative tori, Proceedings of the Seventh Great Plains Operator Theory Seminar (Lawrence, KS, 1987), 1990, pp. 389–397. MR 1065837, DOI
  • F. G. Frobenius, Theorie der linearen Formen mit ganzen coefficienten, J. Reine Angew. Math. 86 (1880), 96-116.
  • Morris Newman, Integral matrices, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 45. MR 0340283

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Keywords: Module, principal ideal domain, pfaffian, skew symmetric matrix, automorphism
Article copyright: © Copyright 1990 American Mathematical Society